A uniform wire of resistance R is shaped into  a regular polygon of n sides where n is even . find the eqivalent resistance between a :-
  1. Opposite corners of the polygon .
  2. Adjacent  corners of the poygon .

Asked by yash.bisht9910 | 3rd Jun, 2015, 04:42: PM

Expert Answer:

begin mathsize 12px style straight A space uniform space wire space of space resistance space straight R space is space shaped space into space space straight a space regular space polygon space of space straight n space sides. The space resistance space of space each space side space of space polygon equals straight R over straight n left parenthesis 1 right parenthesis space Opposite space corners space of space the space polygon space. Across space opposite space corners space of space the space polygon comma space opposite space sides space have space straight n over 2 space side space each. Let space straight R subscript 1 space and space straight R subscript 2 space be space resistances space of space two space parts space of space the space polygon. space So comma space straight R subscript 1 equals space straight R subscript 2 equals straight n over 2 cross times straight R over straight n equals straight R over 2 As space two space parts space are space parallel space to space each space other comma space the space reisistance space of space the space polygon space between space opposite space corners comma straight R apostrophe equals fraction numerator straight R subscript 1 cross times straight R subscript 2 over denominator straight R subscript 1 plus straight R subscript 2 end fraction equals fraction numerator straight R over 2 cross times straight R over 2 over denominator straight R over 2 plus straight R over 2 end fraction equals straight R over 4 left parenthesis 2 right parenthesis space Adjacent space space corners space of space the space poygon space. Let space straight R subscript 1 space be space the space resistance space of space one space side space equals straight R over straight n. straight R subscript 2 space be space the space resistance space of space left parenthesis straight n minus 1 right parenthesis space sides space are space in space series space equals left parenthesis straight n minus 1 right parenthesis straight R over straight n Since space straight R subscript 1 space and space straight R subscript 2 space two space parts space are space in space parallel comma space Hence comma space equivalent space resistance space of space the space polygon space between space the space adjecent space corners comma straight R apostrophe apostrophe equals fraction numerator straight R over straight n cross times left parenthesis straight n minus 1 right parenthesis straight R over straight n over denominator straight R over straight n plus left parenthesis straight n minus 1 right parenthesis straight R over straight n end fraction equals fraction numerator left parenthesis straight n minus 1 right parenthesis straight R over denominator straight n squared end fraction end style

Answered by Priyanka Kumbhar | 4th Jun, 2015, 10:25: AM